Lattice rational arithmetic; definitions and laws
Define addition and subtraction by the compensator rule:
a/b + c/d = ( a*(d;b) + c*(b;d) ) / lcm(b,d)
a/b - c/d = ( a*(d;b) – c*(b;d) ) / lcm(b,d)
Define multiplication, reciprocal and division the usual ways:
(a/b)*(c/d) = (a*c) / (b*d)
1/(c/d) = (d/c)
(a/b)/(c/d) = (a*d)/(b*d)
From these we can define “reduction”, a.k.a. “reciprocal addition”:
(a/b) 1/+ (c/d) = 1 / ( (1/(a/b) ) + (1/(c/d)) )
= lcm(a,c) / ( b*(a;d) + c*(d;a) )
Reduction is like addition, with the roles of numerator and denominator reversed. Therefore they follow similar rules. The following are provable:
(a/A) + (b/B) + (c/C) = ( a (lcm(B,C);A) + b (lcm(C,A);B) + c (lcm(A,B);C) ) / lcm(A,B,C)
(a/A) 1/+ (b/B)1/+ (c/C) = lcm(a,b,c) / ( A (lcm(b,c);a) + B (lcm(c,a);b) + C (lcm(a,b);c) )
Proof requires these lemmas:
(A;B)*(C; lcm(A,B)) = (lcm(A,C) ; B)
( lcm(db, dc) ; da ) = ( lcm(b,c) ; a )
Addition and reduction are commutative.
Proof by symmetry of definitions.
Addition and reduction are associative.
Another proof of symmetry of definitions.
Multiplication triple-distributes over addition and reduction.
Proof involves triple-distribution of multiplication over lcm.
Division distributes from the left, and anti-distributes from the right: (trivial proof)
(A+B+C) / D = (A/D) + (B/D) + (C/D)
(A 1/+ B 1/+C ) / D = (A/D) 1/+ (B/D) 1/+ (C/D)
A / (B+C+D) = (A/B) 1/+ (A/C) 1/+ (A/D)
A / (B 1/+ C 1/+ D) = (A/B) + (A/C) + (A/D)
Identities (trivial proof):
(a/b) = (a/b) + (0/1) = (a/b) 1/+ (1/0) = (a/b) * (1/1)
(a/b) + (-a/b) = 0/b
(a/b) 1/+ (a/-b) = a/0
(a/b) * (b/a) = (ab)/(ab) = 1 + 0/(ab)
x/x = 1 + 0/x
0/-1 is known as the “alternator” @; its reciprocal is -1/0, negative infinity.
a/b + @ = -a/-b = a/b 1/+ - 1/0
Because of them we must weaken distribution to triple distribution:
w(x+y+z) = wx + wy + wz if w is finite
w(x 1/+ y 1/+ z) = wx 1/+ wy 1/+ wz if w is nonzeroid
w(x+y) = wx + wy + 0w if w is finite
w(x 1/+ y ) = wx 1/+ wy 1/+ w/0 if w is nonzeroid
0x * 0y = 0xy
0x + 0y = 0(x+y) = 0(lcm(x,y))
0/(nm) = 0/n + 0/(nm)
(nm)/0 = n/0 1/+ (nm)/0
0/0 is “indefinity”, the indefinite ratio. It is an attractor for addition, reduction, multiplication and division for all finite nonzeroids:
a/b + 0/0 = a/b 1/+ 0/0 = a/b * 0/0 = (a/b)/(0/0) = (0/0)/(a/b) = 0/0
if a and b are both not zero.
Indefinity acts like a generic finite nonzeroid, even though it is both an infinity and a zeroid!
0/0 is an identity for adding infinities, and an attractor for reducing infinities:
a/0 + 0/0 = a/b
a/0 + -a/0 = 0/0
a/0 1/+ 0/0 = 0/0
0/0 is an identity for reducing zeroids, and an attractor for adding zeroids:
0/a 1/+ 0/0 = a/b
0/a 1/+ 0/-a = 0/0
0/a + 0/0 = 0/0